论文标题
带有图色的样品协方差矩阵的微量矩
Trace Moments of the Sample Covariance Matrix with Graph-Coloring
论文作者
论文摘要
令$ s_ {p，n} $表示基于$ n $独立分布的$ n $ p $ dimensional随机向量的样本协方差矩阵。本文的主要结果是，对于高维数据和低维数据，$ s_ {p，n} $的痕量矩和功率跟踪协方差的明确扩展。为此，我们扩展了一个众所周知的Ansatz，将痕量矩描述为路由或图形上的加权总和。我们方法的新颖性是对所检查图的固有着色，以及将图表分解为其树结构和\ textit {seed Graphs}，它允许一些优雅的公式解释了树结构对欧拉旅行数量的影响。图表上的加权总和在可能的种子图上变为加权总和，这反过来又更容易分析。
Let $S_{p,n}$ denote the sample covariance matrix based on $n$ independent identically distributed $p$-dimensional random vectors in the null-case. The main result of this paper is an explicit expansion of trace moments and power-trace covariances of $S_{p,n}$ simultaneously for both high- and low-dimensional data. To this end we expand a well-known ansatz of describing trace moments as weighted sums over routes or graphs. The novelty to our approach is an inherent coloring of the examined graphs and a decomposition of graphs into their tree-structure and their \textit{seed graphs}, which allows for some elegant formulas explaining the effect of the tree structures on the number of Euler-tours. The weighted sums over graphs become weighted sums over the possible seed graphs, which in turn are much easier to analyze.
